From: AAAI Technical Report SS-92-01. Compilation copyright © 1992, AAAI (www.aaai.org). All rights reserved.
An Extended Abstract:
Constraint-Satisfaction
Steven
Minton 1 and
Mark
D. Johnston
A Heuristic
Repair Method for
and Scheduling
Problems
2 and
Andrew
B. Philips
1Sterling FederalSystems
2SpaceTelescope Science Institute
NASAAmes Research Center
3700 San Martin Drive,
AI Research Branch
Baltimore, MD21218 USA
Mail Stop: 269-2
Moffett Field, CA94035 USA
Introduction
One of the most promising general approaches for solving combinatorial search problems is to generate an
initial,
suboptimal solution and then to apply local
repair heuristics. Techniques based on this approach
have met with empirical success on many combinatorial problems, including the traveling salesman and
graph partitioning problems[10]. Such techniques also
have a long tradition in AI, most notably in problemsolving systems that operate by debugging initial solutions [18, 20]. In this paper, we describe how this
idea can be extended to constraint satisfaction problems (CSPs) in a natural manner(see also [14] for full
paper).
Most of the previous work on CSP algorithms has
assumed a standard backtracking approach in which
a partial assignment to the variables is incrementally
extended. In contrast, our method starts with a complete, but inconsistent assignment and then incrementally repairs constraint violations until a consistent
assignment is achieved. The method is guided by a
simple ordering heuristic for repairing constraint violations: identify a variable that is currently in conflict
and select a new value that minimizes the number of
outstanding constraint violations.
Wepresent empirical evidence showing that on some
standard problems our approach is considerably more
efficient than traditional backtracking methods. For
example, on the n-queens problem, our method quickly
finds solutions to the one million queens problem[15].
Weargue that the reason that repair-based methods
can outperform backtracking methods is because a
complete assignment can be more informative in guiding search than a partial assignment. However, the
utility of the extra information is domain dependent.
The work described in this paper was inspired by
a surprisingly effective neural network developed by
Adorfand Johnston [1] for scheduling astronomical observations on the Hubble Space Telescope. Our heuristic CSPmethod was distilled from an analysis of the
network. In the process of carrying out the analysis,
we discovered that the effectiveness of the network has
little to do with its connectionist implementation. Fur-
126
1 3and
Philip
Laird
SNASAAmes Research Center
AI Research Branch
Mail Stop: 269-2
Motfett Field, CA94035 USA
thermore, the ideas employed in the network can be
implemented very efficiently
within a symbolic CSP
framework. The symbolic implementation is extremely
simple. It also has the advantage that several different
search strategies can be employed, although we have
found that hill-climbing methodsare particularly wellsuited for the applications that we have investigated.
Webegin the paper with a brief review of Adorf and
Johnston’s neural network. Following this, we describe
our symbolic methodfor heuristic repair.
Previous
Work:
The GDS Network
By almost any measure, the Hubble Space Telescope
scheduling problem is a complex task [11, 17]. Between ten thousand and thirty thousand astronomical observations per year must be scheduled, subject
to a great variety of constraints including power restrictions, observation priorities, time-dependent orbital characteristics,
movementof astronomical bodies, stray light sources, etc. Because the telescope
is an extremely valuable resource with a limited lifetime, efficient scheduling is a critical concern. Aninitial scheduling system, developed using traditional programming methods, highlighted the difficulty of the
problem; it was estimated that it would take over three
weeks for the system to schedule one week of observations. This problem was remedied by the development
of a successful constraint-based system to augment the
initial system. At the heart of the constraint-based system is a neural network developed by Adorf and Johnston, the Guarded Discrete Stochastic (GDS) network,
which searches for a schedule[l].
From a computational point of view, the network is
interesting because Adorf and Johnston found that it
performs well on a variety of tasks, in addition to the
space telescope scheduling problem. For example, the
network performed significantly better on the n-queens
problem than methods that had been previously developed. The n-queens problem requires placing n queens
on an n × n chessboard so that no two queens share a
row, column or diagonal. The network has been used
to solve problems of up to 1024 queens, whereas most
heuristic backtracking methods encounter difficulties
with problems one-tenth that size[19].
The GDSnetwork is a modified Hopfield network[8].
In a standard Hopfield network, all connections between neurons are symmetric. In the GDSnetwork, the
main network is coupled asymmetrically to an auxiliary
network of guard neurons which restricts the configurations that the network can assume. This modification enables the network to rapidly find a solution for
many problems, even when it is simulated on a serial
machine. Unfortunately, convergence to a stable configuration is no longer guaranteed. Thus the network
can fall into a local minimuminvolving a group of unstable states amongwhich it will oscillate. In practice,
however, if the network fails to converge after some
number of neuron state transitions, it can simply be
stopped and started over.
To solve the n-queens problem with the GDSnetwork, each of the n × n board positions is represented
by a neuron whose output is either one or zero depending on whether or not a queen is located in that position. (Note that this is a local representation rather
than a distributed representation of the board.) If
two board positions are inconsistent, then an inhibiting connection exists between the corresponding two
neurons. For example, all the neurons in a columnwill
inhibit each other, representing the constraint that two
queens cannot be in the same column. For each row,
a guard neuron is connected to each of the neurons in
the row and gives the neurons in that row a large excitatory input, large enough so that at least one neuron
in the row will turn on. Thus, the guard neurons enforce the constraint that one queen in each row must be
on. The network is updated on each cycle by randomly
picking a row and flipping the state of the neuron in
that row whose input is most inconsistent with its current output. A solution is realized when the output of
every neuron is consistent with its input.
Why does the GDS Net Perform
So Well?
Our analysis of the GDSnetwork was motivated by
the question: "Whydoes the network perform so much
better than traditional backtracking methods on certain tasks?" In particular, we were intrigued by the
results on the n-queens problem, since this problem
has received considerable attention from previous researchers. For n-queens, Adorf and Johnston found
empirically that the network requires a linear number
of transitions to converge. Since each transition requires linear time, the expected (empirical) time for
the network to find a solution is O(n2). To check this
behavior, Johnston and Adorf ran experiments with n
as high as 1024, at which point memorylimitations
became a problem.1
1The network, which is programmedin Lisp, requires
approximately 11 minutes to solve the 1024 queens problem on a TI Explorer II. For larger problems, memorybecomesa limiting factor because the the network requires
approximately O(n2) space.
Nonsystematlc
Search Hypothesis
Initially,
we hypothesized that it was the nonsystematic nature of the network’s search that allowed it to
perform muchbetter than systematic depth-first backtracking search. There are two potential problems
associated with systematic depth-first search. First,
the search space may be organized in such a way
that poorer choices are explored first at each branch
point. For instance, in the n-queens problem, depthfirst search tends to find a solution muchmore quickly
whenthe first queen is placed in the center of the first
row rather than the corner. It would appear that solution density is muchgreater in the former case[19], but
most naive algorithms tend to start in the corner simply because humans find it more natural to program
that way. However, the fact that a systematic algorithm may consistantly
make poor choices does not
completely explain why the GDSnetwork performs so
well for n-queens. A backtracking program that randomly orders rows (and columns within rows) performs
muchbetter than the naive method, and yet still performs poorly relative to the GDSnetwork.
The second potential problem with depth-first search
is more significant and more subtle. Depth-first search
can be a disadvantage when solutions are not evenly
distributed throughout the search space. As the distribution of solutions becomes less uniform, and, therefore, the solutions become more clustered, the time
to search between solution clusters increases. Thus,
we conclude that, in a tree where the solutions are
clustered, depth-first search performs relatively poorly.
In comparison, a search strategy which examines the
leaves of the tree in random order is not affected by
solution clustering.
Weinvestigated whether this phenomenonexplained
the relatively poor performance of depth-first search on
n-queens by experimenting with a randomized search
algorithm, called a Las Vegas algorithm [2]. The algorithm begins by selecting a path from the root to
a leaf. To select a path, the algorithm starts at the
root node and chooses one of its children with equal
probability. This process continues recursively until a
leaf is encountered. If the leaf is a solution the algorithm terminates, if not, it starts over again at the
root and selects a path. The same path may be examined more than once, since no memoryis maintained
between successive trials.
The Las Vegas algorithm does, in fact, perform better than simple depth-first search on n-queens. In fact,
this result was already known[2]. However,the performanceof the Las Vegas algorithm is still not nearly as
good as that of the GDSnetwork, and so we concluded
that the systematicity hypothesis alone cannot explain
the network’s behavior.
Informedness
Hypothesis
Our second hypothesis was that the network’s search
process uses information about the current assignment
127
that is not available to a standard backtracking program. Wenow believe this hypothesis is correct, in
that it explains whythe network works so well. In particular, the key to the network’s performance appears
to be that state transitions are madeso as to reduce the
number of outstanding inconsistencies in the network;
specifically, each state transition involves flipping the
neuron whose output is most inconsistent with its current input. Froma constraint satisfaction perspective,
it is as if the networkreassigns a value for a variable by
choosing the value that violates the fewest constraints.
This idea is captured by the following heuristic:
Min-Conflicts heuristic:
Giuen:A set of variables, a set of binary constraints,
and an assignment specifying a value for each variable. Twovariables conflict if their values violate a
constraint.
Procedure:Select a variable that is in conflict, and assign 2it a valuethat minimizesthe number
of conflicts.
(Breakties randomly.)
Wehave found that the network’s behavior can be
approximated by a symbolic system that uses the minconflicts heuristic for hill-climbing. The hill-climbing
system starts with an initial assignment generated in a
preprocessing phase. 3 At each choice point, the heuristic chooses a variable that is currently in conflict and
reassigns its value, until a solution is found. The system thus searches the space of possible assignments,
favoring assignments with fewer total conflicts. Of
course, the hill-climbing system can become "stuck"
in a local maximum,in the same way that the network
may become "stuck" in a local minimum.
There are two aspects of the min-conflicts hillclimbing method that distinguish it from standard
backtracking approaches for CSPproblems. First, instead of extending a consistent partial assignment, the
min-conflicts method repairs a complete but inconsistent assignment by reducing those inconsistencies.
Thus, to guide its search, it uses information about
the current assignment that is not available to a standard backtracking algorithm. Second, the use of a hillclimbing strategy produces a different style of search.
Wehave also found that extracting the method from
the network enables us to tease apart and experiment
with its different components.In particular, the idea of
repairing an inconsistent assignment can be used with
a variety of different search strategies in addition to
hill-climbing.
2In general, the heuristic attempts to minimizethe number of other variables that will need to be repaired. For
binary CSPs, this corresponds to minimizing the number
of conflicting variables. For general CSPs,where a single
constraint mayinvolve several variables, the exact method
of counting the numberof variables that will need to be
repaired depends on the particular constraint. The space
telescope scheduling problemis a general CSP,whereasthe
other tasks described in this paper arc binary CSPs.
3See[14] for an analysis of howdifferent initial assignmentscan affect the repair phase.
128
Highlights
of Experimental
Results
This section contains highlights from experiments in
which we evaluate the performance of the min-conflicts
heuristic on some standard tasks. These experiments
identify problemson which re_in-conflicts performs well,
as well as problems on which it performs poorly. The
experiments also show the extent to which the minconflicts approach approximates the behavior of the
GDS network.
The N-Queens Problem
¯ Min-conflicts
hill-climbing
approximates
the GDS
networkforn-queens.
¯ Forn _> 100rain-conflicts
hill-climbing
hasnever
failed
to finda solution.
¯ Formin-conficts,
therequired
numberof repairs
appearsto remain constan~ as n increases, and the time
to find a solution grows linearly with n.
¯ Standard backtracking using the "most-constrained
first" heuristic quickly grows large: for 100 runs
when n > 1000 a backtracking program implementing the heuristic took more than 12 hours to complete.
¯ Min-conflicts hill-climbing solves the million queens
problem in less than four minutes on a SPAttCstation I.
N-queens is actually quite an easy problem given the
right method.
Scheduling Applications:
HST
¯ Min-conflicts hill-climbing approximates the GDS
network for HSTscheduling.
¯ Muchof the overhead (particularly the space overhead) in the GDSnetwork is eliminated by using the
min-conflicts method.
¯ Because the min-conflicts heuristic is so simple, a
rain-conflicts scheduler for HSTwas quickly coded
in C and is extremely efficient.
¯ The simplicity of the rain-conflicts methodmakes it
easy to experiment with modifications to the heuristic and the search-strategy.
¯ Other telescope scheduling problems have started to
use the min-conflicts scheduler developed for HST.
Graph Coloring
¯ Min-conflicts hill-climbing approximates the GDS
network for graph coloring.
¯ A standard backtracking algorithm employing a
Brelaz-like[3] heuristic outperforms rain-conflicts
hill-climbing.
Summary
of Experimental
Results
For each of the three tasks we have examinedin detail,
n-queens, HSTscheduling and graph 3-colorability, we
have found that the GDSnetwork’s behavior can be
approximated by the rain-conflicts hill-climbing algorithm. To this extent, we have a theory that explains the network’s behavior. Obviously, there are
certain practical advantages to having "extracted" this
method from the network. First, the method is very
simple, and so can be programmed extremely efficiently, especially if done in a task-specific manner.
Second, the heuristic we have identified, that is, choosing the repair which minimizes the numberof conflicts,
is very general. It can be used in combination with different search strategies and task-specific heuristics, an
important factor for most practical applications.
Insofar as the power of our approach is concerned,
our experimental results are encouraging. We have
identified two tasks, n-queens and HSTscheduling,
which appear more amenable to our repair-based approach than a traditional approach that incrementally
extends a partial assignment. This is not to say that
a repair-based approach will do better than any traditional approach for solving these tasks, but merely
that our simple, repair-based method has done relatively well in comparison to the standard traditional
methods. Wealso note that repair-based methods have
a special advantage for scheduling tasks: they can easily be used for both overconstrained and rescheduling
problems. Thus it seems likely that there are other
applications for which our approach will prove useful
Theproblems
investigated
in thispaper,especially
theHST and n-queens
problem,
tendto be relatively
uniform
in thatcritical
constraints
rarelyoccur.In
part,thisis due to the waytheproblems
arerepresented.
Forexample,
in theHSTproblem,
as described
earlier,
thetransitive
closure
of temporal
constraints
is explicitly
represented.
Thus,a single"after"
relationcanbe transformed
intoa setof "after"
relations.
Thisimproves
performance
because
the rain-conflicts
heuristic
is lesslikely
to violate
a setof constraints
thana singleconstraint.
In somecases,we expect
thatmoresophisticated
techniques
willbe necessary
toidentify
critical
constraints[5].
Tothisend,weare
currently
evaluating
explanation-based
learning
techniques
[4,13]asa method
foridentifying
critical
constraints.
Thealgorithms
described
in thispaperalsohavean
important
relation
to previous
workin AI.In particular,thereis a longhistory
of AI programs
thatuse
repair
or debugging
strategies
to solveproblems,
primarily
in theareasofplanning
anddesign[18,
20].This
approach
hasrecently
hada renaissance
withtheemergenceof case-based[6] and analogical [12, 21] problem
solving. To solve a problem, a case-based system will
retrieve the solution from a previous, similar problem
and repair the old solution so that it solves the new
problem.
The fact that the rain-conflicts
approach perDiscussion
forms well on n-queens, a well-studied,
"standard"
The heuristic method described in this paper can be
constraint-satisfaction
problem, suggeststhatAI
repair-based
approaches
may be moregenerally
usecharacterized as a local search method[10], in that each
repair minimizes the number of conficts for an indifulthanpreviously
thought.
However,
in somecasesit
vidual variable. Local search methods have been apcanbe moretime-consuming
to repaira solution
than
plied to a variety of important problems, often with
to construct
a newonefromscratch.
impressive results. For example, the Kernighan-Lin
Therearemanypossible
extensions
to theworkremethod, perhaps the most successful algorithm for
portedhere,butthreeareparticularly
worthmentionsolving graph-partitioning
problems, repeatedly iming.First,weexpectthatthereareotherapplications
proves a partitioning by swapping the two vertices
forwhichtherain-conflicts
approach
willproveuseful.
that yield the greatest cost differential.
The muchConjunctive
matching,
for example,
is an areawhere
publicized simulated annealing method can also be
preliminary
results
appearpromising.
Thisis particucharacterized as a form of local search[9]. However,
laxlytrueformatching
problems
thatrequire
onlythat
it is well-knownthat the effectiveness of local search
a good partial-match
be computed.Second,we exmethods depends greatly on the particular task.
pectthatthereareinteresting
waysin whichtherainIn fact, it is easy to imagine problems on which
conflicts
heuristic
couldbecombined
withotherheuristhe min-conflicts heuristic will fail. The heuristic is
tics.Finally,
thereis thepossibility
of employing
the
poorly suited for problems with a few highly critical
rain-conflicts
heuristic
withothersearch
techniques.
In
constraints and a large number of less important conthispaper,we considered
onlyoneverybasicmethod,
straints.
For example, consider the problem of conhillclimbing.
However,
sincethenumberof conflicts
structing a four-year course schedule for a university
in an assignment can serve as a heuristic evaluation
student.
We mayhavean initial
schedule
whichsatisfunction, more sophisticated techniques such as bestfiesalmost
alloftheconstraints,
except
thata course
first search are possible candidates for investigation.
scheduled
forthefirst
yearis notactually
offered
that
Another possibility is Tabu search[7], a hill-climbing
year.If thiscourseis a prerequisite
forsubsequent technique that maintains a list of forbidden moves in
courses,
thenmanysignificant
changes
to theschedorder to avoid cycles. Morris[16] has also proposed a
ulemaybe required
beforeit is fixed.In general,
if
hill-climbing method which can break out of local maxrepairing
a constraint
violation
requires
completely
reima by systematically altering the cost function. The
visingthecurrent
assignment,
thentherain-conflicts work by Morris and much of the work on Tabu search
heuristic
willoffer
little
guidance.
bears a close relation to our approach.
129
Conclusions
In this paper we have analyzed a very successful neural
network algorithm and shown that an extremely simple, heuristic search method behaves similarly. Based
on our experience with both the GDSnetwork and
min-confficts hill-climbing, we conclude that the rainconflicts heuristic captures the critical aspects of the
GDSnetwork. In this sense, we have explained why
the network is so effective. Additionally, by isolating
the min-conflicts heuristic from the search strategy, we
distinguished the idea of a repair-based CSP method
from the particular strategy employedto search within
the space of repairs.
Finally, there are several practical implications of
this work. First, the scheduling system for the Hubble Space Telescope, SPIKE, now employs our symbolic method, rather than the network, reducing the
overhead necessary to arrive at a schedule. Second,
and perhaps more importantly, it is easy to experiment
with variations of the symbolic method, which should
facilitate transferring SPIKEto other scheduling applications. Third, by demonstrating that repair-based
methods are applicable to standard constraint satisfaction problems, such as N-queens, we have provided a
new tool for solving CSPproblems.
References
[1] H.M. Adoff and M.D. Johnston.
A discrete
stochastic neural network algorithm for constraint
satisfaction problems. In Proceedings of the International Joint Conference on Neural Networks,
San Diego, CA, 1990.
[2] G. Brassard and P. Bratley. Algorithmics - Theory
and Practice. Prentice Hall, EnglewoodCliffs, NJ,
1988.
[3] D. Brelas. Newmethods to color the vertices of a
graph. Communications of the A CM, 22:251-256,
1979.
[4] M. Eskey and M. Zweben. Learning search control for constraint-based scheduling. In Proceedings AAAI-gO, Boston, Mass, 1990.
[5] M.S. Fox, N. Sadeh, and C. Baykan. Constrained
heuristic search. In Proceedings IJCAI.89, Detroit, MI, 1989.
[6] K.J. Hammond.Case-based Planning: An Integrated Theory of Planning, Learning and Mere.
or//. PhDthesis, Yale University, Department of
Computer Science, 1986.
[7] A. Hertz and D. de Werra. Using tabu search
techniques for graph coloring. Computing, 39:345351, 1987.
[8] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. In Proceedings of the National Academyof
Sciences, volume 79, 1982.
[9] D.S. Johnson, C.R. Aragon, L.A. McGeoch, and
C. Schevon. Optimization by simulated annealing:
An experimental evaluation, Part II. To appear in
Journal of Operations Research, 1990.
[10] D.S. Johnson, C.H. Papadimitrou, and M. Yannakakis. How easy is local search? Journal of
Computer and System Sciences, 37:79-100, 1988.
[11] M.D. Johnston. Automated telescope scheduling.
In Proceedings of the Symposium on Coordination
of Observational Projects. Cambridge University
Press, 1987.
[12] S. Kambhampati. Supporting flexible plan reuse.
In Minton S., editor, Machine Learning Methods
for Planning and Scheduling. Morgan Kaufmann,
1992.
[13] S. Minton. Empirical results concerning the utility of explanation-based learning. In Proceedings
AAAI-88, Minneapolis, MN, 1988.
and
[14] S. Minton, M. Johnston, A.B. Philips,
P. Laird. Minimizing conflicts: A heuristic repair
method for constraint-satisfaction
and scheduling
problems. Submitted to Special Issue of Artificial Intelligence Journal on Constraint Satisfaction Problems.
and
[15] S. Minton, M. Johnston, A.B. Philips,
P. Laird. Solving large scale constraint satisfaction and scheduling problems using a heuristic repair method. In Proceedings AAAI-gO, 1990.
[16] P. Morris. Solutions without exhaustive search:
An iterative descent method for binary constraint
satisfaction problems. In Proceedings the AAAI90 Workshop on Constraint-Directed Reasoning,
Boston, MA, 1990.
[17] N. Muscettola,
S.F. Smith, G. Amiri, and
D. Pathak. Generating space telescope observation schedules. Technical Report CMU-RI-TR89-28, Carnegie Mellon University, Robotics Institute, 1989.
[18] R.G. Simmons. A theory of debugging plans and
interpretations.
In Proceedings AAAI-88, Minneapolis, MN,1988.
[19] H.S. Stone and J.M. Stone. Efficient search techniques - an empirical study of the n-queens problem. IBM Journal of Research and Development,
31:464-474, 1987.
[20] G. J. Sussman. A Computer Model of Skill Acquisition. American Elsevier, NewYork, 1975.
[21] M.M.Veloso and J.G. Carbonell. Towards scaling
up machine learning: A case study with derivation
analogy in prodigy. In Minton S., editor, Machine
Learning Methods for Planning and Scheduling.
Morgan Kaufmann, 1992.
130